FOM: Wiener's theorem; Grothendieck universes

[Stephen G Simpson]

Martin Davis 19 Apr 1999 11:06:04 mentions a beautiful theorem of
Wiener, the original hard analysis proof by Wiener, and the subsequent
soft proof by Gelfand using Banach algebra methods.  If I remember
correctly, this is called Wiener's Tauberian theorem.  It's an
important theorem in analysis.  It's also a famous and interesting
case study in soft proofs versus hard proofs.  Rudin's book ``Real and
Complex Analysis'' is a good reference for both aspects.

However, I would not compare this to what is going on with
Grothendieck universes.  My impression is that the technology for
eliminating Grothendieck universes is routine, simply a matter of
cutting things down to size at appropriate places.  McLarty has
exaggerated the difficulty of this.  There is no remotely similar
routine procedure that will take you from Gelfand's soft proof to
Wiener's original hard proof.  (However, the Shoenfield absoluteness
theorem can be used to eliminate the use of AC from Gelfand's proof.)

----

Bill Tait 19 Apr 1999 17:19:59 says 

 > PS. Thank you for not once mentioning universes.

FOM subscribers who are not interested in the Grothendieck universes
thread should feel free to ignore it.

----

Colin McLarty 20 Apr 1999 14:14:06 presents some verbatim quotes from
Grothendieck.  My French is not very good, but it's good enough to see
that this stuff is interesting, at least sociologically.  It's a shame
that Grothendieck and his disciples never published it.

Basically, Grothendieck seems to be whining about how the number
theorists have neglected his beloved Grothendieck topos, replacing it
by commonplace tools of calculation and forgetting their connection to
``the vast All'', a vision which is the sole source of their sense and
power.  He even attacks Bourbaki for failing to incorporate
Grothendieck toposes into the Bourbakian foundational scheme, thereby
giving up the Bourbakian ambition to provide *the* foundation for all
of contemporary mathematics.

Poor Grothendieck.  Is this why he dropped out of mathematics?

F.o.m. amateurism?

At any rate, so far as I can tell, Grothendieck never suggests that
what the number theorists came up with is mathematically meaningless.

A technical question: 

What happens if we replace Grothendieck toposes by elementary toposes?
Does this lead to a recasting of Grothendieck's general theorems about
derived functor cohomology, thereby eliminating Grothendieck universes
while keeping the applications to number theory intact?

My thought is that this might be yet another routine way to eliminate
Grothendieck universes.  

Of course it may not satisfy the religious impulse ....

-- Steve